In particular, the functions \(B,F\)  and \(G\)  in REF  are linear and time-independent.
Thus, it follows from [1]} and [2]} that the PDE (REF ) admits a solution \(V\)  which is Lipschitz continuous in its second variable.
Therefore, if for each \(N\)  the finite player game admits a Nash equilibrium, (see e.g. [3]} for details on the existence) \(\underline{\hat{\alpha }} = (\hat{\alpha }^{1,N},\dots \hat{\alpha }^{N,N})\)  then it follows by Theorem REF  that if \(T\)  is small enough, then the sequence \((L^N(\underline{X}^{\underline{\hat{\alpha }}}))\)  satisfies the LDP with rate function given by (REF ), wherein
\(\overline{B}(t,x, \mu ) = (a+ q) \Big ( \int _{\mathbb {R}}z\mu (dz) - x \Big ) - V(t, x,\mu ).\) 
